The old view that the electron is a zero or one-dimensional point in an all-space probability wave function Ψ(x) is not taken for granted. Rather, atomic and molecular physics theory, derived from first principles, must successfully and consistently apply physical laws on all scales [1-13]. Stability to radiation was ignored by all past atomic models, but in this case, it is the basis of the solutions wherein the structure of the electron is first solved and the result determines the nature of the atomic and molecular electrons involved in chemical bonds. Historically, the point at which quantum mechanics broke with classical laws can be traced to the issue of nonradiation of the one electron atom. Bohr just postulated orbits stable to radiation with the further postulate that the bound electron of the hydrogen atom does not obey Maxwell's equations—rather it obeys different physics [1-13]. Later physics was replaced by “pure mathematics” based on the notion of the inexplicable wave-particle duality nature of electrons which lead to the Schrödinger equation wherein the consequences of radiation predicted by Maxwell's equations were ignored. Ironically, Bohr, Schrödinger, and Dirac used the Coulomb potential, and Dirac used the vector potential of Maxwell's equations. But, all ignored electrodynamics and the corresponding radiative consequences. Dirac originally attempted to solve the bound electron physically with stability with respect to radiation according to Maxwell's equations with the further constraints that it was relativistically invariant and gave rise to electron spin [14]. He and many founders of QM such as Sommerfeld, Bohm, and Weinstein wrongly pursued a planetary model, were unsuccessful, and resorted to the current mathematical-probability-wave model that has many problems [1-18]. Consequently, Feynman for example, attempted to use first principles including Maxwell's equations to discover new physics to replace quantum mechanics [19].
Starting with the same essential physics as Bohr, Schrödinger, and Dirac of e− moving in the Coulombic field of the proton and an electromagnetic wave equation and matching electron source current rather than an energy diffusion equation originally sought by Schrödinger, advancements in the understanding of the stability of the bound electron to radiation are applied to solve for the exact nature of the electron. Rather than using the postulated Schrödinger boundary condition: “Ψ→0 as r→∞”, which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell's equations, the structure of the electron is derived as a boundary-value problem wherein the electron comprises the source current of time-varying electromagnetic fields during transitions with the constraint that the bound n=1 state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution modeled as a superposition of accelerating charges does not have to radiate. The physical boundary condition of nonradiation of that was imposed on the bound electron follows from a derivation by Haus [20]. The function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light. Similarly, nonradiation is demonstrated based on the electron's electromagnetic fields and the Poynting power vector. A simple invariant physical model arises naturally wherein the results are extremely straightforward, internally consistent, and predictive of conjugate parameters for the first time, requiring minimal math as in the case of the most famous exact equations (no uncertainty) of Newton and Maxwell on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used.
The structure of the bound atomic electron was solved by first considering one-electron atoms [1-13]. Since the hydrogen atom is stable and nonradiative, the electron has constant energy. Furthermore, it is time dynamic with a corresponding current that serves as a source of electromagnetic radiation during transitions. The wave equation solutions of the radiation fields permit the source currents to be determined as a boundary-value problem. These source currents match the field solutions of the wave equation for two dimensions plus time when the nonradiation condition is applied. Then, the mechanics of the electron can be solved from the two-dimensional wave equation plus time in the form of an energy equation wherein it provides for conservation of energy and angular momentum as given in the Electron Mechanics and the Corresponding Classical Wave Equation for the Derivation of the Rotational Parameters of the Electron section of Ref. [1]. Once the nature of the electron is solved, all problems involving electrons can be solved in principle. Thus, in the case of one-electron atoms, the electron radius, binding energy, and other parameters are solved after solving for the nature of the bound electron.
For time-varying spherical electromagnetic fields, Jackson [21] gives a generalized expansion in vector spherical waves that are convenient for electromagnetic boundary-value problems possessing spherical symmetry properties and for analyzing multipole radiation from a localized source distribution. The Green function G(x′,x) which is appropriate to the equation(∇2+k2)G(x′,x)=−δ(x′−x)  (1)in the infinite domain with the spherical wave expansion for the outgoing wave Green function is
                              G          ⁡                      (                                          x                ′                            ,              x                        )                          =                                            ⅇ                                                -                  ⅈ                                ⁢                                                                  ⁢                k                ⁢                                                                        x                    -                                          x                      ′                                                                                                                            4              ⁢              π              ⁢                                                                x                  -                                      x                    ′                                                                                                ⁢                                          ⁢                                          =                      i            ⁢                                                  ⁢            k            ⁢                                          ∑                                  l                  =                  0                                ∞                            ⁢                                                                    j                    l                                    ⁡                                      (                                          kr                      <                                        )                                                  ⁢                                                      h                    l                                          (                      1                      )                                                        ⁡                                      (                                          kr                      <                                        )                                                  ⁢                                                      ∑                                          m                      =                                              -                        l                                                              l                                    ⁢                                                                                    Y                                                  l                          ,                          m                                                *                                            ⁡                                              (                                                                              θ                            ′                                                    ,                                                      ϕ                            ′                                                                          )                                                              ⁢                                                                  Y                                                  l                          ,                          m                                                                    ⁡                                              (                                                  θ                          ,                          ϕ                                                )                                                                                                                                                    (        2        )            Jackson [21] further gives the general multipole field solution to Maxwell's equations in a source-free region of empty space with the assumption of a time dependence eiωnt:
                              B          =                                    ∑                              l                ,                m                                      ⁢                          [                                                                                          a                      E                                        ⁡                                          (                                              l                        ,                        m                                            )                                                        ⁢                                                            f                      l                                        ⁡                                          (                      kr                      )                                                        ⁢                                      X                                          l                      ,                      m                                                                      -                                                      i                    k                                    ⁢                                                            a                      M                                        ⁡                                          (                                              l                        ,                        m                                            )                                                        ⁢                                      ∇                                          ×                                                                        g                          l                                                ⁡                                                  (                          kr                          )                                                                    ⁢                                              X                                                  l                          ,                          m                                                                                                                                ]                                      ⁢                                  ⁢                  E          =                                    ∑                              l                ,                m                                      ⁢                          [                                                                    i                    k                                    ⁢                                                            a                      E                                        ⁡                                          (                                              l                        ,                        m                                            )                                                        ⁢                                      ∇                                          ×                                                                        f                          l                                                ⁡                                                  (                          kr                          )                                                                    ⁢                                              X                                                  l                          ,                          m                                                                                                                    +                                                                            a                      M                                        ⁡                                          (                                              l                        ,                        m                                            )                                                        ⁢                                                            g                      l                                        ⁡                                          (                      kr                      )                                                        ⁢                                      X                                          l                      ,                      m                                                                                  ]                                                          (        3        )            where the cgs units used by Jackson are retained in this section. The radial functions ft(kr) and gt(kr) are of the form:gt(kr)=At(1)ht(1)+At(2)ht(2)  (4)Xt.m is the vector spherical harmonic defined by
                                                        X                              l                ,                m                                      ⁡                          (                              θ                ,                ϕ                            )                                =                                    1                                                l                  ⁡                                      (                                          l                      +                      1                                        )                                                                        ⁢                                          LY                                  l                  ,                  m                                            ⁡                              (                                  θ                  ,                  ϕ                                )                                                    ⁢                                  ⁢        where                            (        5        )                                L        =                              1            i                    ⁢                      (                          r              ×              ∇                        )                                              (        6        )            The coefficients aE(l,m) and aM(l,m) of Eq. (3) specify the amounts of electric (l,m) multipole and magnetic (l,m) multipole fields, and are determined by sources and boundary conditions as are the relative proportions in Eq. (4). Jackson gives the result of the electric and magnetic coefficients from the sources as
                                                        a              E                        ⁡                          (                              l                ,                m                            )                                =                                                    4                ⁢                π                ⁢                                                                  ⁢                                  k                  2                                                            i                ⁢                                                      l                    ⁡                                          (                                              l                        +                        1                                            )                                                                                            ⁢                          ∫                                                Y                  l                                      m                    *                                                  ⁢                                  {                                                                                                                                                                                                                                          ρ                                  ⁢                                                                                                                                                    ∂                                                                                                                                                                                                                                              ∂                                        r                                                                                                              ⁡                                                                          [                                                                                                                        rj                                          l                                                                                ⁡                                                                                  (                                          kr                                          )                                                                                                                    ]                                                                                                                                      +                                                                                                                                                                                                                                                                              ik                                    c                                                                    ⁢                                                                      (                                                                          r                                      ·                                      J                                                                        )                                                                    ⁢                                                                                                            j                                      l                                                                        ⁡                                                                          (                                      kr                                      )                                                                                                                                      -                                                                                                                                                                                                                                                  ik                          ⁢                                                      ∇                                                          ·                                                              (                                                                  r                                  ×                                  M                                                                )                                                                                                              ⁢                                                                                    j                              l                                                        ⁡                                                          (                              kr                              )                                                                                                                                                            }                                ⁢                                                      ⅆ                    3                                    ⁢                  x                                                                    ⁢                                  ⁢        and                            (        7        )                                                      a            M                    ⁡                      (                          l              ,              m                        )                          =                                                            -                4                            ⁢              π              ⁢                                                          ⁢                              k                2                                                                    l                ⁡                                  (                                      l                    +                    1                                    )                                                              ⁢                      ∫                                                            j                  l                                ⁡                                  (                  kr                  )                                            ⁢                              Y                l                                  m                  *                                            ⁢                              L                ·                                  (                                                            J                      c                                        +                                          ∇                                              ×                        M                                                                              )                                            ⁢                                                ⅆ                  3                                ⁢                x                                                                        (        8        )            respectively, where the distribution of charge ρ(x,t), current J(x,t), and intrinsic magnetization M(x,t) are harmonically varying sources: ρ(x)e−iωt, J(x)e−iωt, and M(x)e−iωt.
The electron current-density function can be solved as a boundary value problem regarding the time varying corresponding source current J(x)e−iωt that gives rise to the time-varying spherical electromagnetic fields during transitions between states with the further constraint that the electron is nonradiative in a state defined as the n=1 state. The potential energy, V(r), is an inverse-radius-squared relationship given by given by Gauss' law which for a point charge or a two-dimensional spherical shell at a distance r from the nucleus the potential is
                              V          ⁡                      (            r            )                          =                  -                                    ⅇ              2                                      4              ⁢                              πɛ                0                            ⁢              r                                                          (        9        )            Thus, consideration of conservation of energy would require that the electron radius must be fixed. Addition constraints requiring a two-dimensional source current of fixed radius are matching the delta function of Eq. (1) with no singularity, no time dependence and consequently no radiation, absence of self-interaction (See Appendix III of Ref. [1]), and exact electroneutrality of the hydrogen atom wherein the electric field is given by
                              n          ·                      (                                          E                1                            -                              E                2                                      )                          =                              σ            s                                ɛ            0                                              (        10        )            where n is the normal unit vector, E1 and E2 are the electric field vectors that are discontinuous at the opposite surfaces, σx is the discontinuous two-dimensional surface charge density, and E2=0. Then, the solution for the radial electron function, which satisfies the boundary conditions is a delta function in spherical coordinates—a spherical shell [22]
                              f          ⁡                      (            r            )                          =                              1                          r              2                                ⁢                      δ            ⁡                          (                              r                -                                  r                  n                                            )                                                          (        11        )            where rn is an allowed radius. This function defines the charge density on a spherical shell of a fixed radius (See FIG. 1), not yet determined, with the charge motion confined to the two-dimensional spherical surface. The integer subscript n is determined during photon absorption as given in the Excited States of the One-Electron Atom (Quantization) section of Ref. [1]. It is shown in this section that the force balance between the electric fields of the electron and proton plus any resonantly absorbed photons gives the result that rn=nr1 wherein n is an integer in an excited state.
Given time harmonic motion and a radial delta function, the relationship between an allowed radius and the electron wavelength is given by2πrn=λn  (12)Based on conservation of the electron's angular momentum of, the magnitude of the velocity and the angular frequency for every point on the surface of the bound electron are
                              v          n                =                              h                                          m                e                            ⁢                              λ                n                                              =                                    h                                                m                  e                                ⁢                2                ⁢                π                ⁢                                                                  ⁢                                  r                  n                                                      =                                                            m                  e                                ⁢                                  r                  n                                                                                        (        13        )                                          ω          n                =                                            m              e                        ⁢                          r              n              2                                                          (        14        )            To further match the required multipole electromagnetic fields between transitions of states, the trial nonradiative source current functions are time and spherical harmonics, each having an exact radius and an exact energy. Then, each allowed electron charge-density (mass-density) function is the product of a radial delta function
      (                  f        ⁡                  (          r          )                    =                        1                      r            2                          ⁢                  δ          ⁡                      (                          r              -                              r                n                                      )                                )    ,two angular functions (spherical harmonic functions Ytm(θ,φ)=Plm(cos θ)eimφ), and a time-harmonic function eimωnt. The spherical harmonic Y00(θ,φ)=1 is also an allowed solution that is in fact required in order for the electron charge and mass densities to be positive definite and to give rise to the phenomena of electron spin. The real parts of the spherical harmonics vary between −1 and 1. But the mass of the electron cannot be negative; and the charge cannot be positive. Thus, to insure that the function is positive definite, the form of the angular solution must be a superposition:Y00(θ,φ)+Ylm(θ,φ)  (15)The current is constant at every point on the surface for the s orbital corresponding to Y00(θ,φ). The quantum numbers of the spherical harmonic currents can be related to the observed electron orbital angular momentum states. The currents corresponding to s, p, d, f, etc. orbitals are
                              l          =          0                ⁢                                  ⁢                                  ⁢                              ρ            ⁡                          (                              r                ,                θ                ,                ϕ                ,                t                            )                                =                                                    ⅇ                                  8                  ⁢                  π                  ⁢                                                                          ⁢                                      r                    2                                                              ⁡                              [                                  δ                  ⁡                                      (                                          r                      -                                              r                        n                                                              )                                                  ]                                      ⁡                          [                                                                    Y                    0                    0                                    ⁡                                      (                                          θ                      ,                      ϕ                                        )                                                  +                                                      Y                    l                    m                                    ⁡                                      (                                          θ                      ,                      ϕ                                        )                                                              ]                                                          (        16        )                                          l          ≠          0                ⁢                                  ⁢                                  ⁢                              ρ            ⁡                          (                              r                ,                θ                ,                ϕ                ,                t                            )                                =                                                    ⅇ                                  4                  ⁢                  π                  ⁢                                                                          ⁢                                      r                    2                                                              ⁡                              [                                  δ                  ⁡                                      (                                          r                      -                                              r                        n                                                              )                                                  ]                                      ⁡                          [                                                                                                                                            Y                          0                          0                                                ⁡                                                  (                                                      θ                            ,                            ϕ                                                    )                                                                    +                                                                                                                                  Re                      ⁢                                              {                                                                                                            Y                              l                              m                                                        ⁡                                                          (                                                              θ                                ,                                ϕ                                                            )                                                                                ⁢                                                      ⅇ                                                          ⅈ                              ⁢                                                                                                                          ⁢                                                              mω                                n                                                            ⁢                              t                                                                                                      }                                                                                                        ]                                                          (        17        )            where Ylm(θ,φ) are the spherical harmonic functions that spin about the z-axis with angular frequency ωn with Y00(θ,φ) the constant function and Re{Ylm(θ,φ)einωnt}=Plm(cos θ)cos(mφ+mωnt).
The Fourier transform of the electron charge-density function is a solution of the four-dimensional wave equation in frequency space (k, ω-space). Then the corresponding Fourier transform of the current-density function K(s,Θ,Φ,ω) is given by multiplying by the constant angular frequency corresponding to a potentially emitted photon.
                              K          ⁡                      (                          s              ,              Θ              ,              Φ              ,              ω                        )                          =                  4          ⁢                                          ⁢                      πω            n                    ⁢                                                    sin                ⁡                                  (                                      2                    ⁢                                          s                      n                                        ⁢                                          r                      n                                                        )                                                            2                ⁢                                  s                  n                                ⁢                                  r                  n                                                      ⊗            2                    ⁢          π          ⁢                                                    ∑                                  υ                  =                  1                                            ∞                        ⁢                                                                                                      (                                              -                        1                                            )                                                              υ                      -                      1                                                        ⁢                                                            (                                              π                        ⁢                                                                                                  ⁢                        sin                        ⁢                                                                                                  ⁢                        Θ                                            )                                                              2                      ⁢                                              (                                                  υ                          -                          1                                                )                                                                                                                                                        (                                              υ                        -                        1                                            )                                        !                                    ⁢                                                            (                                              υ                        -                        1                                            )                                        !                                                              ⁢                                                                    Γ                    ⁡                                          (                                              1                        2                                            )                                                        ⁢                                      Γ                    ⁡                                          (                                              υ                        +                                                  1                          2                                                                    )                                                                                                                                  (                                              π                        ⁢                                                                                                  ⁢                        cos                        ⁢                                                                                                  ⁢                        Θ                                            )                                                                                      2                        ⁢                                                                                                  ⁢                        υ                                            +                      1                                                        ⁢                                      2                                          υ                      +                      1                                                                                  ⁢                                                2                  ⁢                                      υ                    !                                                                                        (                                          υ                      -                      1                                        )                                    !                                            ⁢                                                s                                                            -                      2                                        ⁢                    υ                                                  ⊗                2                            ⁢              π              ⁢                                                ∑                                      υ                    =                    1                                    ∞                                ⁢                                                                                                                              (                                                      -                            1                                                    )                                                                          υ                          -                          1                                                                    ⁢                                                                        (                                                      π                            ⁢                                                                                                                  ⁢                            sin                            ⁢                                                                                                                  ⁢                            Φ                                                    )                                                                          2                          ⁢                                                      (                                                          υ                              -                              1                                                        )                                                                                                                                                                                        (                                                      υ                            -                            1                                                    )                                                !                                            ⁢                                                                        (                                                      υ                            -                            1                                                    )                                                !                                                                              ⁢                                                                                    Γ                        ⁡                                                  (                                                      1                            2                                                    )                                                                    ⁢                                              Γ                        ⁡                                                  (                                                      υ                            +                                                          1                              2                                                                                )                                                                                                                                                              (                                                      π                            ⁢                                                                                                                  ⁢                            cos                            ⁢                                                                                                                  ⁢                            Φ                                                    )                                                                                                      2                            ⁢                                                                                                                  ⁢                            υ                                                    +                          1                                                                    ⁢                                              2                                                  υ                          +                          1                                                                                                      ⁢                                                            2                      ⁢                                              υ                        !                                                                                                            (                                                  υ                          -                          1                                                )                                            !                                                        ⁢                                      s                                                                  -                        2                                            ⁢                      υ                                                        ⁢                                                            1                                              4                        ⁢                        π                                                              ⁡                                          [                                                                        δ                          ⁡                                                      (                                                          ω                              -                                                              ω                                n                                                                                      )                                                                          +                                                  δ                          ⁡                                                      (                                                          ω                              +                                                              ω                                n                                                                                      )                                                                                              ]                                                                                                                              (        18        )            The motion on the orbitsphere is angular; however, a radial correction exists due to special relativistic effects. When the velocity is c corresponding to a potentially emitted photonsn·vn=sn·c=ωn  (19)the relativistically corrected wavelength is (Eq. (1.247) of Ref. [1])rn=λn  (20)Substitution of Eq. (20) into the sinc function results in the vanishing of the entire Fourier transform of the current-density function. Thus, spacetime harmonics of
            ω      n        c    =            k      ⁢                          ⁢      or      ⁢                          ⁢                        ω          n                c            ⁢                        ɛ                      ɛ            o                                =    k  for which the Fourier transform of the current-density function is nonzero do not exist. Radiation due to charge motion does not occur in any medium when this boundary condition is met. There is acceleration without radiation. (Also see Abbott and Griffiths and Goedecke [23-24]). Nonradiation is also shown directly using Maxwell's equations directly in Appendix 1 of Ref. [1]. However, in the case that such a state arises as an excited state by photon absorption, it is radiative due to a radial dipole term in its current-density function since it possesses spacetime Fourier transform components synchronous with waves traveling at the speed of light as shown in the Instability of Excited States section of Ref. [1]. The radiation emitted or absorbed during electron transitions is the multipole radiation given by Eq. (2) as given in the Excited States of the One-Electron Atom (Quantization) section and the Equation of the Photon section of Ref. [1] wherein Eqs. (4.18-4.23) give a macro-spherical wave in the far-field.
In Chapter 1 of Ref. [1], the uniform current density function Y00(θ,φ) (Eqs. (16-17)) that gives rise to the spin of the electron is generated from two current-vector fields (CVFs). Each CVF comprises a continuum of correlated orthogonal great circle current-density elements (one dimensional “current loops”). The current pattern comprising each CVF is generated over a half-sphere surface by a set of rotations of two orthogonal great circle current loops that serve as basis elements about each of the
                    (                              -                          i              x                                ,                      i            y                    ,                      0            ⁢                          i              z                                      )            ⁢                          ⁢      and      ⁢                          ⁢              (                                            -                              1                                  2                                                      ⁢                          i              x                                ,                                    1                              2                                      ⁢                          i              y                                ,                      i            z                          )              -    axis    ;the span being π radians. Then, the two CVFs are convoluted, and the result is normalized to exactly generate the continuous uniform electron current density function Y00(θ,φ) covering a spherical shell and having the three angular momentum components of
      L    xy    =                    +                  /                      -                          ℏ              4                                          ⁢              (                              +                          /                              -                designates                                              ⁢                                          ⁢          both          ⁢                                          ⁢          the          ⁢                                          ⁢          positive          ⁢                                          ⁢          and          ⁢                                          ⁢          negative          ⁢                                          ⁢          vector          ⁢                                          ⁢          directions          ⁢                                          ⁢          along          ⁢                                          ⁢          an          ⁢                                          ⁢          axis          ⁢                                          ⁢          in          ⁢                                          ⁢          the          ⁢                                          ⁢          xy          ⁢                      -                    ⁢          plane                )            ⁢                          ⁢      and      ⁢                          ⁢              L        z              =                  ℏ        2            .      The z-axis view of a representation of the total current pattern of the Y00(θ,φ) orbitsphere comprising the superposition of 144 current elements is shown in FIG. 2A. As the number of great circles goes to infinity the current distribution becomes continuous and is exactly uniform following normalization. A representation of the
      (                            -                      1                          2                                      ⁢                  i          x                    ,                        1                      2                          ⁢                  i          y                    ,              i        z              )    ⁢      -    ⁢  axisview of the total uniform current-density pattern of the Y00(φ,θ) orbitsphere with 144 vectors overlaid on the continuous bound-electron current density giving the direction of the current of each great circle element is shown in FIG. 2B. This superconducting current pattern is confined to two spatial dimensions.
Thus, a bound electron is a constant two-dimensional spherical surface of charge (zero thickness and total charge=−e), called an electron orbitsphere that can exist in a bound state at only specified distances from the nucleus determined by an energy minimum for the n=1 state and integer multiples of this radius due to the action of resonant photons as shown in the Determination of Orbitsphere Radii section and Excited States of the One-Electron Atom (Quantization) section of Ref. [1], respectively. The bound electron is not a point, but it is point-like (behaves like a point at the origin). The free electron is continuous with the bound electron as it is ionized and is also point-like as shown in the Electron in Free Space section of Ref. [1]. The total function that describes the spinning motion of each electron orbitsphere is composed of two functions. One function, the spin function (see FIG. 1 for the charge function and FIG. 2 for the current function), is spatially uniform over the orbitsphere, where each point moves on the surface with the same quantized angular and linear velocity, and gives rise to spin angular momentum. It corresponds to the nonradiative n=1, l=0 state of atomic hydrogen which is well known as an s state or orbital. The other function, the modulation function, can be spatially uniform—in which case there is no orbital angular momentum and the magnetic moment of the electron orbitsphere is one Bohr magneton—or not spatially uniform—in which case there is orbital angular momentum. The modulation function rotates with a quantized angular velocity about a specific (by convention) z-axis. The constant spin function that is modulated by a time and spherical harmonic function as given by Eq. (17) is shown in FIG. 3 for several t values. The modulation or traveling charge-density wave that corresponds to an orbital angular momentum in addition to a spin angular momentum are typically referred to as p, d, f, etc. orbitals and correspond to an l quantum number not equal to zero.
It was shown previously [1-13] that classical physics gives closed form solutions for the atom including the stability of the n=1 state and the instability of the excited states, the equation of the photon and electron in excited states, the equation of the free electron, and photon which predict the wave particle duality behavior of particles and light. The current and charge density functions of the electron may be directly physically interpreted. For example, spin angular momentum results from the motion of negatively charged mass moving systematically, and the equation for angular momentum, r×p, can be applied directly to the wavefunction (a current density function) that describes the electron. The magnetic moment of a Bohr magneton, Stern Gerlach experiment, g factor, Lamb shift, resonant line width and shape, selection rules, correspondence principle, wave-particle duality, excited states, reduced mass, rotational energies, and momenta, orbital and spin splitting, spin-orbital coupling, Knight shift, and spin-nuclear coupling, and elastic electron scattering from helium atoms, are derived in closed form equations based on Maxwell's equations. The agreement between observations and predictions based on closed-form equations with fundamental constants only matches to the limit permitted by the error in the measured fundamental constants.
In contrast to the failure of the Bohr theory and the nonphysical, unpredictive, adjustable-parameter approach of quantum mechanics, multielectron atoms [1, 5] and the nature of the chemical bond [1, 6] are given by exact closed-form solutions containing fundamental constants only. Using the nonradiative electron current-density functions, the radii are determined from the force balance of the electric, magnetic, and centrifugal forces that correspond to the minimum of energy of the atomic or ionic system. The ionization energies are then given by the electric and magnetic energies at these radii. The spreadsheets to calculate the energies from exact solutions of one through twenty-electron atoms are available from the internet [25]. For 400 atoms and ions the agreement between the predicted and experimental results are remarkable [5]. Here I extend these results to the nature of the chemical bond. In this regard, quantum mechanics has historically sought the lowest energy of the molecular system, but this is trivially the case of the electrons inside the nuclei. Obviously, the electrons must obey additional physical laws since matter does not exist in a state with the electrons collapsed into the nuclei. Specifically, molecular bonding is due to the physics of Newton's and Maxwell's laws together with achieving an energy minimum.
The structure of the bound molecular electron was solved by first considering the one-electron molecule H2+ and then the simplest molecule H2 [1, 6]. The nature of the chemical bond was solved in the same fashion as that of the bound atomic electron. First principles including stability to radiation requires that the electron charge of the molecular orbital is a prolate spheroid, a solution of the Laplacian as an equipotential minimum energy surface in the natural ellipsoidal coordinates compared to spheroidal in the atomic case, and the current is time harmonic and obeys Newton's laws of mechanics in the central field of the nuclei at the foci of the spheroid. There is no a priori reason why the electron position must be a solution of the three-dimensional wave equation plus time and cannot comprise source currents of electromagnetic waves that are solutions of the three-dimensional wave equation plus time. Then, the special case of nonradiation determines that the current functions are confined to two-spatial dimensions plus time and match the electromagnetic wave-equation solutions for these dimensions. In addition to the important result of stability to radiation, several more very important physical results are subsequently realized: (i) The charge is distributed on a two-dimension surface; thus, there are no infinities in the corresponding fields (Eq. (10)). Infinite fields are simply renormalized in the case of the point-particles of quantum mechanics, but it is physically gratifying that none arise in this case since infinite fields have never been measured or realized in the laboratory. (ii) The hydrogen molecular ion or molecule has finite dimensions rather than extending over all space. From measurements of the resistivity of hydrogen as a function of pressure, the finite dimensions of the hydrogen molecule are evident in the plateau of the resistivity versus pressure curve of metallic hydrogen [26]. This is in contradiction to the predictions of quantum probability functions such as an exponential radial distribution in space. Furthermore, despite the predictions of quantum mechanics that preclude the imaging of a molecule orbital, the full three-dimensional structure of the outer molecular orbital of N2 has been recently tomographically reconstructed [27]. The charge-density surface observed is similar to that shown in FIG. 4 for H2 which is direct evidence that MO's electrons are not point-particle probability waves that have no form until they are “collapsed to a point” by measurement. Rather they are physical, two-dimensional equipotential charge density functions as derived herein. (iii) Consistent with experiments, neutral scattering is predicted without violation of special relativity and causality wherein a point must be everywhere at once as required in the QM case. (iv) There is no electron self-interaction. The continuous charge-density function is a two-dimensional equipotential energy surface with an electric field that is strictly normal for the elliptic parameter ξ>0 according to Gauss' law and Faraday's law. The relationship between the electric field equation and the electron source charge-density function is given by Maxwell's equation in two dimensions [28,29] (Eq. (10)). This relation shows that only a two-dimensional geometry meets the criterion for a fundamental particle. This is the nonsingularity geometry that is no longer divisible. It is the dimension from which it is not possible to lower dimensionality. In this case, there is no electrostatic self-interaction since the corresponding potential is continuous across the surface according to Faraday's law in the electrostatic limit, and the field is discontinuous, normal to the charge according to Gauss' law [28-30]. (v) The instability of electron-electron repulsion of molecular hydrogen is eliminated since the central field of the hydrogen molecular ion relative to a second electron at ξ>0 which binds to form the hydrogen molecule is that of a single charge at the foci. (vi) The ellipsoidal MOs allow exact spin pairing over all time that is consistent with experimental observation. This aspect is not possible in the QM model.
Current algorithms to solve molecules are based on nonphysical models based on the concept that the electron is a zero or one-dimensional point in an all-space probability wave function Ψ(x) that permits the electron to be over all space simultaneously and give output based on trial and error or direct empirical adjustment of parameters. These models ultimately cannot be the actual description of a physical electron in that they inherently violate physical laws. They suffer from the same shortcomings that plague atomic quantum theory, infinities, instability with respect to radiation according to Maxwell's equations, violation of conservation of linear and angular momentum, lack of physical relativistic invariance, and the electron is unbounded such that the edge of molecules does not exist. There is no uniqueness, as exemplified by the average of 150 internally inconsistent programs per molecule for each of the 788 molecules posted on the NIST website [31]. Furthermore, from a physical perspective, the implication for the basis of the chemical bond according to quantum mechanics being the exchange integral and the requirement of zero-point vibration, “strictly quantum mechanical phenomena,” is that the theory cannot be a correct description of reality as described for even the simple bond of molecular hydrogen as reported previous [1, 6]. Even the premise that “electron overlap” is responsible for bonding is opposite to the physical reality that negative charges repel each other with an inverse-distance-squared force dependence that becomes infinite. A proposed solution based on physical laws and fully compliant with Maxwell's equations solves the parameters of molecules even to infinite length and complexity in closed form equations with fundamental constants only.
For the first time in history, the key building blocks of organic chemistry have been solved from two basic equations. Now, the true physical structure and parameters of an infinite number of organic molecules up to infinite length and complexity can be obtained to permit the engineering of new pharmaceuticals and materials at the molecular level. The solutions of the basic functional groups of organic chemistry were obtained by using generalized forms of a geometrical and an energy equation for the nature of the H—H bond. The geometrical parameters and total bond energies of about 800 exemplary organic molecules were calculated using the functional group composition. The results obtained essentially instantaneously match the experimental values typically to the limit of measurement [1]. The solved function groups are given in Table 1.
TABLE 1Partial List of Organic FunctionalGroups Solved by Classical Physics.Continuous-Chain AlkanesBranched AlkanesAlkenesBranched AlkenesAlkynesAlkyl FluoridesAlkyl ChloridesAlkyl BromidesAlkyl IodidesAlkenyl HalidesAryl HalidesAlcoholsEthersPrimary AminesSecondary AminesTertiary AminesAldehydesKetonesCarboxylic AcidsCarboxylic Acid EstersAmidesN-alkyl AmidesN,N-dialkyl AmidesUreaCarboxylic Acid HalidesCarboxylic Acid AnhydridesNitrilesThiolsSulfidesDisulfidesSulfoxidesSulfonesSulfitesSulfatesNitroalkanesAlkyl NitratesAlkyl NitritesConjugated AlkenesConjugated PolyenesAromaticsNapthaleneTolueneChlorobenzenePhenolAnilineAryl Nitro CompoundsBenzoic Acid CompoundsAnisolePyrroleFuranThiopheneImidizolePyridinePyrimidinePyrazineQuinolineIsoquinolineIndoleAdenineFullerene (C60)GraphitePhosphinesPhosphine OxidesPhosphitesPhosphates
The two basic equations that solves organic molecules, one for geometrical parameters and the other for energy parameters, were applied to bulk forms of matter containing trillions of trillions of electrons. For example, using the same alkane- and alkene-bond solutions as elements in an infinite network, the nature of the solid molecular bond for all known allotropes of carbon (graphite, diamond, C60, and their combinations) were solved. By further extension of this modular approach, the solid molecular bond of silicon and the nature of semiconductor bond were solved. The nature of other fundamental forms of matter such as the nature of the ionic bond, the metallic bond, and additional major fields of chemistry such as that of silicon, organometallics, and boron were solved exactly such that the position and energy of each and every electron is precisely specified. The implication of these results is that it is possible using physical laws to solve the structure of all types of matter. Some of the solved forms of matter of infinite extent as well as additional major fields of chemistry are given in Table 2. In all cases, the agreement with experiment is remarkable [1].
TABLE 2Partial List of Additional Molecules and Compositionsof Matter Solved by Classical Physics.Solid Molecular Bond of the Three Allotropes of CarbonDiamondGraphiteFullerene (C60)Solid Ionic Bond of Alkali-HydridesAlkali-Hydride Crystal StructuresLithium HydrideSodium HydridePotassium HydrideRubidium & Cesium HydridePotassium Hydrino HydrideSolid Metallic Bond of Alkali MetalsAlkali Metal Crystal StructuresLithium MetalSodium MetalPotassium MetalRubidium & Cesium MetalsAlkyl Aluminum HydridesSilicon Groups and MoleculesSilanesAlkyl Silanes and DisilanesSolid Semiconductor Bond of SiliconInsulator-Type Semiconductor BondConductor-Type Semiconductor BondBoron MoleculesBoranesBridging Bonds of BoranesAlkoxy BoranesAlkyl BoranesAlkyl Borinic AcidsTertiary AminoboranesQuaternary AminoboranesBorane AminesHalido Boranes Organometallic MolecularFunctional Groups and MoleculesAlkyl Aluminum HydridesBridging Bonds of Organoaluminum HydridesOrganogermanium and DigermaniumOrganoleadOrganoarsenicOrganoantimonyOrganobismuthOrganic Ions1° Amino2° AminoCarboxylatePhosphateNitrateSulfateSilicateProteinsAmino AcidsPeptide BondsDNABases2-deoxyriboseRibosePhosphate Backbone
The background theory of classical physics (CP) for the physical solutions of atoms and atomic ions is disclosed in Mills journal publications [1-13], R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2000 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'00 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, September 2001 Edition, BlackLight Power, Inc., Cranbury, N.J., Distributed by Amazon.com (“'01 Mills OUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, July 2004 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'04 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2005 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'05 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. L. Mills, “The Grand Unified Theory of Classical Quantum Mechanics”, June 2006 Edition, Cadmus Professional Communications-Science Press Division, Ephrata, Pa., ISBN 0963517171, Library of Congress Control Number 2005936834, (“'06 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, October 2007 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'07 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Physics, June 2008 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'08 Mills GUT-CP”); in prior published PCT applications WO2005/067678; WO2005/116630; WO2007/051078; WO2007/053486; and WO2008/085,804, and U.S. Pat. No. 7,188,033; U.S. Application No. 60/878,055, filed 3 Jan. 2007; 60/880,061, filed 12 Jan. 2007; 60/898,415, filed 31 Jan. 2007; 60/904,164, filed 1 Mar. 2007; 60/907,433, filed 2 Apr. 2007; 60/907,722, filed 13 Apr. 2007; 60/913,556, filed 24 Apr. 2007; 60/986,675, filed 9 Nov. 2007; 60/988,537, filed 16 Nov. 2007; 61/018,595, filed 2 Jan. 2008; 61/027,977, filed 12 Feb. 2008; 61/029,712, filed 19 Feb. 2008; and 61/082,701, filed 22 Jul. 22, 2008, the entire disclosures of which are all incorporated herein by reference (hereinafter “Mills Prior Publications”).